yd. qr. na. Hence, From 5 0 0 Difference, 4 3 311 4. From of 5 acres of land, subtract of 3 roods. Ans. 2R. 44P. 5. From of an ounce, take of a pennyweight. Ans. 7pwt. 15gr. 6. From of a hogshead, take of a quart. 94. Ans. 6gal. 3qt. kpt. EXERCISES IN DENOMINATE FRACTIONS. 1. A person gave for some thread, and did he pay for all? of a pound for a hat, of a shilling of a penny for a needle. What Ans. 3s. 2d. 34far. 2. What is the value of of a week, of a day, and of a minute? Ans. 3da. 20hr. 15sec. 3. What is the value of of a pound, of an ounce, and of a pennyweight, Troy? it per pound? Ans. 2oz. 13pwt. 34gr. cents, how much is Ans. 10 cents. 5. If I pay $4.04 for 84 bushels of apples, how much do I give per bushel? Ans. 46 cents. 6. Four persons, A, B, C, and D, own a ship, of which A owns of of the whole; B owns of as much as A; C owns as much as B; and D owns the remainder. What are the respective parts owned by each ? 7. From of of a day of 24 hours, take of of 14 Ans. 8h. 50m. hour. 8. To of 4 days of 24 hours each, add 1 of 1 of 31 hours. Ans. 3d. 9d. 11m. 40sec. 9. A certain sum of money is to be divided between 4 persons, in such a manner that the first shall have of it, the second, the third, and the fourth the remainder, which is $28. What is the sum? ++, which wants just of being the whole; hence, the fourth one had of the whole. Consequently, $28 is of the whole, and the whole is $28×4=$112. 10. A received of a legacy, B, and C the remainder. Now it is found that A had $80 more than B. How much did each receive? Hence, $80 was of the whole legacy; the legacy was therefore $80×15=$1200. Hence, A had of $1200-$200. Proof, $1200, 11. Eight detachments of artillery divided 4608 cannon balls in the following manner: The first took 72 and ¦ of the remainder; the second took 144 and of the remainder; the third took 216 and of the remainder; the fourth took 288 and of the remainder. The balance was equally divided among the remaining four detachments, How many balls did each detachment receive? Ans. Each received 576 balls. 12. Five persons divide 100 pounds of sugar as follows. The first takes of of the whole; the second takes / of of the remainder; the third takes of of the remainder; the fourth takes of of the remainder ; and the fifth had what was left. How much did each receive? lb. lb. lb. oz. dr. of 100=10 11 6. 66 2d had 100% of 10025 of 100=11 294. The 1st had 1536 of 1002 3d had 10 of 100 of 100=11 11 8. 14336 4th had 15 of 100=255 of 100=12 7 31. 14336 5th had 7735 of 1000 of 100=53 15 41. 14336 VULGAR FRACTIONS REDUCED TO DECIMALS. 95. To change a vulgar fraction into an equivalent decimal fraction. Let us endeavor to change into an equivalent decimal fraction. This fraction is the same as of a unit; and as 10 tenths make a unit, the fraction is the same as 3 of 4o of a tenth, 3 tenths+ of a tenth. Again, § of a tenth is the same as of 10 of one hundredth,-7 hundredths+ of one hundredth. But of one hundredth is the same as of 10 of one thousandth,=5 thousandths. Therefore of a unit 3 tenths, 7 hundredths, and 5 thousandths, or as usually written, =0·375. Hence we deduce this RULE. Annex a cipher to the numerator, and then divide by the denominator. If the dividend will not contain the divisor, write O in the quotient and annex another cipher, and then divide; to the remainder annex another cipher, and again divide by the denominator; and so continue to do until there is no remainder, or until as many decimal figures have been obtained as may be desired. The quotient will be the decimal fraction required. NOTE. It will be seen that this rule bears a close analogy to rule under ART. 91, as it ought; since the values of the successive figures in a decimal fraction decrease in a tenfold ratio. 3. What decimal is equivalent to? 4. What decimal is equivalent to? 5. What decimal is equivalent to ? 6. What decimal is equivalent to ? Ans. 0.05. Ans. 0.04. Ans. 0.3333, &c. Ans. 0.142857, &c. 7. What decimal is equivalent to? Ans. 0-0909, &c. 8. What decimal is equivalent to? Ans. 0-076923, &c. 9. What decimal is equivalent to †? Ans. 0.0588235, &c. 10. Change into an equivalent decimal. Ans. 0·76. 16. Change into an equivalent decimal. Ans. 0·875. 17. Change into an equivalent decimal. Ans. 0.95. 13 Ans. 0.928571428, &c. In the foregoing process of converting a vulgar fraction into an equivalent decimal fraction, we continue to annex ciphers to the remainders, and to divide by the denominator of the vulgar fraction; hence, whenever we obtain a remainder like one that has previously occurred, then the decimal figures will commence a repetition. And as no remainder can exceed or equal the divisor or denominator of the vulgar fraction, the whole number of different remainders cannot exceed the number of units in the denominator less one; consequently, when the decimal figures do not terminate, they must recur in periods whose number of places cannot exceed the number of units less one in the denominator of the equivalent vulgar fraction. Decimals which recur in this way, are called repetends. When the period begins with the first decimal figure, it |